In this work, we use the algebra of coupled scalars to develop two kinds of nonlinear integrable couplings of the modified Korteweg-de Vries (mKdV) equation. One of the integrable couplings of the mKdV equation gives multiple soliton solutions of distinct amplitudes, whereas the second kind gives multiple singular soliton solutions of distinct amplitudes as well. The Bäcklund transformation and the simplified Hirota's method will be used for this study. We show that these couplings possess multiple soliton solutions the same as the multiple soliton solutions of the mKdV equation, but differ only in the coefficients of the Bäcklund transformation. This difference exhibits soliton solutions with distinct amplitudes.
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In this work, we present construction of the integrable couplings of the Burgers equation (BE) and the Sharma-Tasso-Olver (TSO) equation. We use the algebra of coupled scalars to develop the two classes of couplings. The B̈acklund transformation and the simplified Hirota's method will be used to study the developed couplings. We show that these couplings possess multiple kink solutions the same as the multiple kink solutions of the BE and the STO equations, but differ only in the coefficients of the B̈acklund transformation. This difference exhibits kink solutions for some equations and anti-kink solutions for others.
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In this paper, we establish the Volterra integro-differential forms of the singular Flierl-Petviashvili equation and the singular Lane-Emden equation. We use the variational iteration method (VIM) to effectively handle any singular equation of the form identical to these equations. The Volterra integro-differential forms of the singular equations overcome the singular behaviour at the origin x = 0, do not use a variety of Lagrange multipliers, and facilitate the computational work. The Padé approximant will be used for the Flierl-Petviashvili equation that is valid in an infinite domain.
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